Week 7: Spatial Linear Models

a) Goals

This worked example shows:

• How to test regression residuals for spatial autocorrelation.
• How to fit a model with spatially autocorrelated errors (GLS).
• How to fit a spatial simultaneous autoregressive error model (SAR).
• How to perform spatial filtering with Moran eigenvector maps (MEM).
• How to fit a spatially varying coefficients model (SVC).

b) Data set

Here we analyze population-level data of the wildflower Dianthus carthusianorum (common name: Carthusian pink) in 65 calcareous grassland patches in the Franconian Jura, Germany (Rico et al. 2013):

• Dianthus: Spatial point data frame with population-level data (patch characteristics, grazing regime, genetic diversity, 15 alternative connectivity indices Si), included in package ‘LandGenCourse’. To load the data, type (without quotes): ‘data(Dianthus)’. For a definition of the variables, type: ‘?Dianthus’.

c) Required R packages

Note: the function ‘library’ will always load the package, even if it is already loaded, whereas ‘require’ will only load it if it is not yet loaded. Either will work.

library(LandGenCourse)
#library(here)
#libraryspdep)
library(nlme)
#library(lattice)
#library(MuMIn)
#library(gridExtra)
library(dplyr)
library(spatialreg)
library(ggplot2)
library(ggmap)
source(system.file("extdata", "panel.cor.r", 
                            package = "LandGenCourse"))

Package ‘spmoran’ not automatically installed with ‘LandGenCourse’:

if(!require(spmoran)) install.packages("spmoran", repos='http://cran.us.r-project.org')
#require(spmoran)

a) Import data

data(Dianthus)

Allelic richness ‘A’ was not calculate for populations with < 5 individuals. Here we extract only the patches with ‘A’ values, and the variables needed, and store them in a data frame ‘Dianthus.df’.

b) Create a map

The functions in package ‘ggmap’ expect the projection to be Longitude/Latitude. We could start by transforming the projection of the Dianthus dataset (ignore the warning about CRS object having no comment):

Dianthus.longlat <- sp::spTransform(Dianthus, 
             CRSobj = sp::CRS(SRS_string = "EPSG:4326"))

## Warning in spTransform(xSP, CRSobj, ...): NULL source CRS comment, falling back
## to PROJ string

## Warning in spTransform(xSP, CRSobj, ...): +init dropped in PROJ string

## Warning in wkt(obj): CRS object has no comment

## Warning in wkt(obj): CRS object has no comment

## Warning in showSRID(uprojargs, format = "PROJ", multiline = "NO", prefer_proj = prefer_proj): Discarded datum Deutsches_Hauptdreiecksnetz in Proj4 definition,
##  but +towgs84= values preserved

make_bbox(lon = Longitude, lat = Latitude, data = Dianthus@data, f = 0.2)

##     left   bottom    right      top 
## 10.91051 48.90282 11.08951 49.07073

Here, however, we use a shortcut as the ‘Dianthus@data’ dataframe already contains variables with Longitude and Latitude. We use these to define the bounding box for the map, extending the map extent by 20% (‘f=0.2’).

bbox <- make_bbox(lon = Longitude, lat = Latitude, data = Dianthus@data, f = 0.2)

Now we can obtain a stamen map for the study area from the internet and add points for the sampling sites. Here we use a terrain map.

Note: here we use two types of maps from the source ‘stamen’, which is freely available: a terrain map and a toner map. Starting in mid 2018, accessing maps from Google with ‘source = google’ requires an API key.

To plot the maps side-by-side, we first write each map into an object, then use the function gridExtra::grid.arrange to plot them on the same row. This works for a wide range of figures, including ggplot2 or trellis-type plots.

StamenMap.terrain <- ggmap(get_stamenmap(bbox, maptype = "terrain", zoom=12)) + 
  geom_point(aes(x =  Longitude, y = Latitude), data = Dianthus.longlat@data, 
             colour = "black", size = 2)

StamenMap.toner <- ggmap::ggmap(get_stamenmap(bbox, maptype = "toner", zoom=12, force = TRUE)) + 
  geom_point(aes(x =  Longitude, y = Latitude), data = Dianthus.longlat@data, 
             colour = "black", size = 2) 

gridExtra::grid.arrange(StamenMap.terrain, StamenMap.toner, nrow=1)

As you can see from the terrain map, most sites lie on the steep slopes between an upper and a lower Jurassic plateau. A few sites lie at the forest edge on the upper plateau, typically in areas where the soil is too shallow to allow crop farming. With in the study area, all known sites were sampled. Additional sites are expected to be found mainly in the valley system in the Southwest.

There are three types of stamen maps: terrain, toner (black and white), and watercolor. Terrain and toner maps have three components each (background, labels, lines), which can be downloaded individually if needed. Hence, the argument ‘maptype’ has the following options:

• terrain
• terrain-background
• terrain-labels
• terrain-lines
• toner
• toner-2010
• toner-background
• toner-hybrid
• toner-labels
• toner-lines
• toner-lite
• watercolor

c) Explore correlations

When fitting linear models, it is always a good idea to look at the correlations first.

Dianthus.df <- data.frame(A=Dianthus@data$A, IBD=Dianthus@data$Eu_pj, 
                          IBR=Dianthus@data$Sheint_pj,
                          PatchSize=log(Dianthus@data$Ha),
                          System=Dianthus@data$System,
                          Longitude=Dianthus@data$Longitude,
                          Latitude=Dianthus@data$Latitude,
                          x=Dianthus@coords[,1], y=Dianthus@coords[,2])

# Define 'System' for ungrazed patches
Dianthus.df$System=as.character(Dianthus@data$System)
Dianthus.df$System[is.na(Dianthus.df$System)] <- "Ungrazed"
Dianthus.df$System <- factor(Dianthus.df$System, 
                             levels=c("Ungrazed", "East", "South", "West"))

# Remove patches with missing values for A
Dianthus.df <- Dianthus.df[!is.na(Dianthus.df$A),]
dim(Dianthus.df)

## [1] 59  9

pairs(Dianthus.df[,-c(5:7)], lower.panel=panel.smooth, 
      upper.panel=panel.cor, diag.panel=panel.hist)

Questions:

• How strong is the association between ‘Eu_pj’ and ‘A’? What does this suggest about the hypothesis of IBD?
• How strong is the association between ‘Sheint_pj’ and ‘A’? What does this suggest about the hypothesis of sheep-mediated gene flow (IBR)?
• Which variable seems to be a better predictor of allelic richness: patch size ‘Ha’ or the logarithm of patch size, ‘PatchSize’?
• Is logHa correlated with ‘IBD’ or ‘IBR’?
• Are any of the variables correlated with the spatial coordinates x and y?

Do the three grazing systems, and the ungrazed patches, differ in allelic richness A? Also, let’s check the association between patch size and population size. Here we create boxplots that show the individual values as dots. We add a horizontal jitter to avoid overlapping points.

Boxplot1 <- ggplot(Dianthus.df, aes(x=System, y=A)) + 
  geom_boxplot() + xlab("Grazing system") + ylab("Allelic richness (A)") +
  geom_jitter(shape=1, position=position_jitter(0.1), col="blue")

Boxplot2 <- ggplot(Dianthus@data, aes(x=factor(pop09), y=log(Ha))) + 
  geom_boxplot() + xlab("Population size class") + ylab("PatchSize (log(Ha))") +
  geom_jitter(shape=1, position=position_jitter(0.1), col="blue")

gridExtra::grid.arrange(Boxplot1, Boxplot2, nrow=1)

Even though the population size categories were very broad, there appears to be a strong relationship between populations size (category) and (the logarithm of) patch size.

Despite this relationship, connectivity models Si that only considered Dianthus carthusianorum presence/absence (‘pj’) in source patches ‘j’ were better supported than those Si models that took into account source patch area (‘Aj’) or population size (‘Nj’).

We can check this by calculating the correlation of allelelic richness ‘A’ with each of the 15 connectivity models ‘Si’ in the data set.

round(matrix(cor(Dianthus@data$A, Dianthus@data[,15:29], 
                 use="pairwise.complete.obs"), 5, 3, byrow=TRUE, 
           dimnames=list(c("Eu", "Shecte", "Sheint", "Shenu", "Forest"), 
                         c("pj", "Aj", "Nj"))),3)

##           pj     Aj    Nj
## Eu     0.024  0.207 0.098
## Shecte 0.375  0.273 0.369
## Sheint 0.403  0.196 0.369
## Shenu  0.135 -0.045 0.105
## Forest 0.137  0.168 0.138

• Correlations with ‘A’ are highest for the two ‘IBR’ models that assume seed dispersal over a limited number of patches along shepherding routes (‘Shecte’ and ‘Sheint’. These two models include only continuously grazed, or both continuously and intermittently grazed patches, respectively.
• Correlations for models that take in to account population size (‘Nj’) are only slightly lower, whereas those that use patch size (‘Aj’) as a proxy for the size of the seed emigrant pool had lower correlations.

a) Fit regression models

Here we fit three multiple regression models to explain variation in allelic richness:

• mod.lm.IBD: IBD model of connectivity ‘Eu_pj’.
• mod.lm.IBR: IBR model shepherding connectivity ‘Sheint_pj’.
• mod.lm.PatchSize: log patch size and IBR model.

mod.lm.IBD <- lm(A ~ IBD, data = Dianthus.df)
summary(mod.lm.IBD)

## 
## Call:
## lm(formula = A ~ IBD, data = Dianthus.df)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.68545 -0.10220  0.03883  0.16178  0.36100 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 4.070960   0.064061  63.549   <2e-16 ***
## IBD         0.008454   0.047460   0.178    0.859    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.2324 on 57 degrees of freedom
## Multiple R-squared:  0.0005563,  Adjusted R-squared:  -0.01698 
## F-statistic: 0.03173 on 1 and 57 DF,  p-value: 0.8593

This model does not fit the data at all!

mod.lm.IBR <- lm(A ~ IBR, data = Dianthus.df)
summary(mod.lm.IBR)

## 
## Call:
## lm(formula = A ~ IBR, data = Dianthus.df)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.66844 -0.11251  0.03418  0.12219  0.41760 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  3.92306    0.05499  71.348  < 2e-16 ***
## IBR          0.25515    0.07672   3.326  0.00155 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.2128 on 57 degrees of freedom
## Multiple R-squared:  0.1625, Adjusted R-squared:  0.1478 
## F-statistic: 11.06 on 1 and 57 DF,  p-value: 0.001547

This model fits much better. Let’s check the residuals plots.

par(mfrow=c(2,2), mar=c(4,4,2,1))
plot(mod.lm.IBR)

par(mfrow=c(1,1))

The residuals show some deviation from a normal distribution. Specifically, the lowest values are lower than expected.

mod.lm.PatchSize <- lm(A ~ PatchSize + IBR, data = Dianthus.df)
summary(mod.lm.PatchSize)

## 
## Call:
## lm(formula = A ~ PatchSize + IBR, data = Dianthus.df)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.68359 -0.08844  0.02538  0.11705  0.41705 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  4.05158    0.07781  52.069   <2e-16 ***
## PatchSize    0.04266    0.01888   2.260   0.0277 *  
## IBR          0.11338    0.09709   1.168   0.2479    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.2055 on 56 degrees of freedom
## Multiple R-squared:  0.2325, Adjusted R-squared:  0.2051 
## F-statistic: 8.482 on 2 and 56 DF,  p-value: 0.0006058

This combinde model explains more variation in allelic richness than the IBR model alone. Moreover, after adding PatchSizes, the IBR term is no longer statistically significant!

Has the distribution of residuals improved as well?

par(mfrow=c(2,2), mar=c(4,4,2,1))
plot(mod.lm.PatchSize)

par(mfrow=c(1,1))

Not really!

b) Test for spatial autocorrelation (Moran’s I):

Before we interpret the models, let’s check whether the assumption of independent residuals is violated by spatial autocorrelation in the residuals.

To calculate and test Moran’s I, we first need to define neighbours and spatial weights. Here we use a Gabriel graph to define neighbours.

We define weights in three ways (see Week 5 video and tutorial for code explanation):

• listw.gab: 1 = neighbour, 0 = not a neighbour.
• listw.d1: inverse distance weights: neighbour j with weight 1/dij
• listw.d2: inverse squared distance weights: neighbour j with weight 1/dij^2

In each case, we row-standardize the weights with the option ‘style = “W”’.

Note: when using ‘graph2nb’, make sure to use the argument ‘sym=TRUE’. This means that if A is a neighbour of B, B is also a neighbour of A. The default is ‘sym=FALSE’, which may result in some sites not having any neighbours assigned (though this would not be evident from the figure!).

xy <- data.matrix(Dianthus.df[,c("x", "y")])
nb.gab <- spdep::graph2nb(spdep::gabrielneigh(xy), sym=TRUE)
par(mar=c(0,0,0,0))
plot(nb.gab, xy)

listw.gab <- spdep::nb2listw(nb.gab)

dlist <- spdep::nbdists(nb.gab, xy)
dlist <- lapply(dlist, function(x) 1/x)
listw.d1 <- spdep::nb2listw(nb.gab, style = "W", glist=dlist)
dlist <- lapply(dlist, function(x) 1/x^2)
listw.d2 <- spdep::nb2listw(nb.gab, style = "W", glist=dlist)

Now we can quantify and test Moran’s I for each variable to test for spatial autocorrelation in response and predictor variables. For now, we’ll take the simple weights ‘listw.gab’.

spdep::moran.test(Dianthus.df$A, listw.gab)             

## 
##  Moran I test under randomisation
## 
## data:  Dianthus.df$A  
## weights: listw.gab    
## 
## Moran I statistic standard deviate = 2.881, p-value = 0.001982
## alternative hypothesis: greater
## sample estimates:
## Moran I statistic       Expectation          Variance 
##        0.29180913       -0.01724138        0.01150691

spdep::moran.test(Dianthus.df$IBD, listw.gab)

## 
##  Moran I test under randomisation
## 
## data:  Dianthus.df$IBD  
## weights: listw.gab    
## 
## Moran I statistic standard deviate = 5.9689, p-value = 1.194e-09
## alternative hypothesis: greater
## sample estimates:
## Moran I statistic       Expectation          Variance 
##        0.62709841       -0.01724138        0.01165320

spdep::moran.test(Dianthus.df$IBR, listw.gab) 

## 
##  Moran I test under randomisation
## 
## data:  Dianthus.df$IBR  
## weights: listw.gab    
## 
## Moran I statistic standard deviate = 3.008, p-value = 0.001315
## alternative hypothesis: greater
## sample estimates:
## Moran I statistic       Expectation          Variance 
##        0.30918167       -0.01724138        0.01177588

spdep::moran.test(Dianthus.df$PatchSize, listw.gab) 

## 
##  Moran I test under randomisation
## 
## data:  Dianthus.df$PatchSize  
## weights: listw.gab    
## 
## Moran I statistic standard deviate = 3.4257, p-value = 0.0003066
## alternative hypothesis: greater
## sample estimates:
## Moran I statistic       Expectation          Variance 
##        0.35490160       -0.01724138        0.01180103

Questions:

• Which variables showed statistically signficant spatial autocorrelation?
• Which variables showed the strongest autocorrelation? Is this surprising?

Next, let’s test each model for autocorrelation in the residuals:

spdep::lm.morantest(mod.lm.IBD, listw.gab) 

## 
##  Global Moran I for regression residuals
## 
## data:  
## model: lm(formula = A ~ IBD, data = Dianthus.df)
## weights: listw.gab
## 
## Moran I statistic standard deviate = 2.9439, p-value = 0.00162
## alternative hypothesis: greater
## sample estimates:
## Observed Moran I      Expectation         Variance 
##       0.28889983      -0.02854559       0.01162751

spdep::lm.morantest(mod.lm.IBR, listw.gab)          

## 
##  Global Moran I for regression residuals
## 
## data:  
## model: lm(formula = A ~ IBR, data = Dianthus.df)
## weights: listw.gab
## 
## Moran I statistic standard deviate = 1.883, p-value = 0.02985
## alternative hypothesis: greater
## sample estimates:
## Observed Moran I      Expectation         Variance 
##       0.18032443      -0.02296810       0.01165614

spdep::lm.morantest(mod.lm.PatchSize, listw.gab)       

## 
##  Global Moran I for regression residuals
## 
## data:  
## model: lm(formula = A ~ PatchSize + IBR, data = Dianthus.df)
## weights: listw.gab
## 
## Moran I statistic standard deviate = 1.7484, p-value = 0.04019
## alternative hypothesis: greater
## sample estimates:
## Observed Moran I      Expectation         Variance 
##       0.16223745      -0.02708922       0.01172530

Quite a bit of the spatial autocorrelation in allelic richness can be explained by the spatial structure in the predictors IBR and PatchSize. There is still statistically significant spatial autocorrelation in the residuals, though it is not strong any more.

a) Plot empirical variogram

The error structure in a GLS is defined in a geostatistical framework, based on a variogram and as a function of distance between observations. Hence we start with plotting an empirical variogram of the residuals, with a smooth line. Here we specify ’resType = “normalized”, which means that the variogram will be fitted to the normalized residuals of the model.

The expected value of the semivariance will be 1. Hence it would make sense to add a horizontal line at 1. However, this is cumbersome with the trellis graphics (using package ‘lattice’) used by ‘nlme’.

model.lm <- nlme::gls(A ~ IBR + PatchSize, data = Dianthus.df, method="REML")
semivario <- nlme::Variogram(model.lm, form = ~x  + y, resType = "normalized")

If you want to create your own figure, e.g. with ‘ggplot2’, you can access the values stores in the data frame ‘semivario’ to plot the points, and add a smooth line yourself. Then we can add a horizontal line with ‘geom_hline’.

ggplot(data=semivario, aes(x=dist, y=variog)) + geom_point() + geom_smooth(se=FALSE) +
  geom_hline(yintercept=1) + ylim(c(0,1.3)) + xlab("Distance") + ylab("Semivariance")

## `geom_smooth()` using method = 'loess' and formula 'y ~ x'

Question:

• What do you conclude from this empirical variogram?
• Estimate the range of the variogram from the intersection of the smooth line with the horizontal line.
• Estimate the nugget effect from the intercept at Distance = 0.

b) Fit variogram models

We can ask R to fit different types of variogram models to this empirical variogram. The model family (e.g., exponential, gaussian, spherical) determines the general shape of the curve that will be fitted. With ‘nugget=T’, we indicate that a nugget effect should be fitted.

Note: Here we want to compare mixed models with the same fixed effects but different random effect structures defined by correlation. For this, we use REML. If we wanted to compare models with the same random effects but different fixed effects (as in Week 6), we should use maximum likelihood.

• With function lme4::lmer, we can set REML=TRUE for REML and REML=FALSE for ML.
• Here with nlme::gls, we set method="REML" for REML and method="ML" for ML.
• For update to work here, you’ll need to load the library nlme(or use this, with three coons to access a function ‘update’, for ‘lme’ objects, that is not ‘exported’ from the package: nlme:::update.lme)

model.lm <- nlme::gls(A ~ IBR + PatchSize, data = Dianthus.df, method="REML")

mod.corExp <- update(model.lm, correlation = nlme::corExp(form = ~ x + y, nugget=T))
mod.corGaus <- update(model.lm, correlation = nlme::corGaus(form = ~ x + y, nugget=T))
mod.corSpher <- update(model.lm, correlation = nlme::corSpher(form = ~ x + y, nugget=T))
mod.corRatio <- update(model.lm, correlation = nlme::corRatio(form = ~ x + y, nugget=T))
#mod.corLin <- update(model.lm, correlation = nlme::corLin(form = ~ x + y, nugget=T))

c) Select best-fitting model

Now we compare all models for which we did not get an error message:

MuMIn::model.sel(model.lm, mod.corExp, mod.corGaus, mod.corSpher, mod.corRatio)     

## Model selection table 
##              (Intrc)     IBR   PtchS             family            correlation
## mod.corExp     4.044 0.08468 0.04269 gaussian(identity)   nlme::corExp(~x+y,T)
## mod.corRatio   4.032 0.09704 0.04129 gaussian(identity) nlme::corRatio(~x+y,T)
## model.lm       4.052 0.11340 0.04266 gaussian(identity)                       
## mod.corGaus    4.047 0.11840 0.04247 gaussian(identity)  nlme::corGaus(~x+y,T)
## mod.corSpher   4.048 0.11710 0.04242 gaussian(identity) nlme::corSpher(~x+y,T)
##              df logLik AICc delta weight
## mod.corExp    6  6.283  1.1  0.00  0.336
## mod.corRatio  6  6.229  1.2  0.11  0.319
## model.lm      4  3.702  1.3  0.29  0.291
## mod.corGaus   6  3.763  6.1  5.04  0.027
## mod.corSpher  6  3.760  6.1  5.04  0.027
## Models ranked by AICc(x)

The list sorts the models, with the best model on top. The last column ‘weight’ contains the model weight, which indicate how much support there is for each model, given all other models in the set (see Week 12). Here, the exponential model fitted best, though the ratio model and the model without a spatially correlated error structure fitted the data almost equally well. The top three models have delta values within 2 (in fact, close to 0).

We refit the best model with maximum likelihood to test the fixed effects.

mod.corExp.ML <- nlme::gls( A ~ PatchSize + IBR, data = Dianthus.df, method="ML",
                            correlation = nlme::corExp(form = ~ x + y, nugget=T))
car::Anova(mod.corExp.ML) 

## Analysis of Deviance Table (Type II tests)
## 
## Response: A
##           Df  Chisq Pr(>Chisq)  
## PatchSize  1 5.4912    0.01911 *
## IBR        1 0.8710    0.35067  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

The fitted model with the exponential error structure shows a significant effect for PatchSize but not for the IBR term.

We don’t get an R-squared value directly, but we can calculate a pseudo R-squared from a regression of the response ‘A’ on the fitted values (using the model fitted with REML). Let’s compare it to the R-squared from the lm model.

summary(lm(A ~ fitted(mod.corExp), data = Dianthus.df))$r.squared

## [1] 0.2317629

summary(mod.lm.PatchSize)$r.squared

## [1] 0.2324922

The pseudo R-squared is almost identical to the R-squared of the non-spatial lm model.

Let’s check the residual plots:

predictmeans::residplot(mod.corExp)

The normal probability plot still looks about the same.

Note that the function residplot recognized that we have a gls model and added a plot of the auto-correlation function, ACF. Here we have a value of 1 for the distance lag 0, which is the comparison of each value with itself. All other values are low. Recall that the autcorrelation is inversely related to the semivariogram (which does not report the value for lag = 0):

semivario <- nlme::Variogram(mod.corExp, form = ~ x + y, 
                             resType = "normalized")
plot(semivario, smooth = TRUE)

The variogram of the residuals (after accounting for spatial autocorrelation as modeled by the variogram model) does look better!

d) Plot fitted variogram model

How can we plot the fitted variogram? Let’s first store it in an object ‘Fitted.variog’, then plot it. Note that the fitted variogram itself has two classes, “Variogram” and “data.frame”. The plot created by plot(Fitted.variog) is a “trellis” object.

Fitted.variog <- nlme::Variogram(mod.corExp)
class(Fitted.variog)

## [1] "Variogram"  "data.frame"

class(plot(Fitted.variog))

## [1] "trellis"

plot(Fitted.variog)

That was easy. However, trellis plots are difficult to tweak, and we may want to create our own plot with ggplot2. For this, we need to access the fitted variogram values (i.e. the exponential model curve values), which is a bit more involved.

The object ‘Fitted.variog’ is a data frame (S3) with additional attributes. This raises a challenge, because we access attributes of S3 objects with $, but we also use $ to access columns in a data frame.

If we just print Fitted.variog, we only see the data frame.

head(Fitted.variog)

##      variog      dist n.pairs
## 1 0.6782704  620.5202      85
## 2 0.6112605 1236.8297      86
## 3 0.7415494 1827.3367      85
## 4 0.5751311 2269.6729      86
## 5 1.0503017 2691.2494      85
## 6 0.7493939 3074.7370      86

We can see the attributes listed by using str:

str(Fitted.variog)

## Classes 'Variogram' and 'data.frame':    20 obs. of  3 variables:
##  $ variog : num  0.678 0.611 0.742 0.575 1.05 ...
##  $ dist   : num  621 1237 1827 2270 2691 ...
##  $ n.pairs: int [1:20(1d)] 85 86 85 86 85 86 85 86 85 86 ...
##  - attr(*, "modelVariog")=Classes 'Variogram' and 'data.frame':  50 obs. of  2 variables:
##   ..$ variog: num [1:50] 0.598 0.637 0.673 0.705 0.734 ...
##   ..$ dist  : num [1:50] 106 385 665 945 1224 ...
##  - attr(*, "collapse")= logi TRUE

The line we are looking for is: - attr(*, "modelVariog"). The attribute modelVariog has 50 rows (obs.) and 2 variables: $variog and $dist. These are the fitted values (i.e., values of the exponential variogram model for 50 distance values).

The notation attr(*, "modelVariog") is a cryptic way of telling us how to access the attribute: use the function attr and provide two arguments: the object names Fitted.variog (represented by the asterisk), and the name of the attribute, in quotes: attr(Fitted.variog, "modelVariog").

tibble::as_tibble(attr(Fitted.variog, "modelVariog"))

## # A tibble: 50 × 2
##    variog  dist
##     <dbl> <dbl>
##  1  0.598  106.
##  2  0.637  385.
##  3  0.673  665.
##  4  0.705  945.
##  5  0.734 1224.
##  6  0.760 1504.
##  7  0.784 1783.
##  8  0.805 2063.
##  9  0.824 2342.
## 10  0.841 2622.
## # … with 40 more rows

This is useful to know if you want to create your own figures, e.g. with ggplot2.

ggplot(data=Fitted.variog, aes(x=dist, y=variog)) + geom_point() + 
  ylim(c(0,1.3)) + xlab("Distance") + ylab("Semivariance") + 
  geom_line(data=attr(Fitted.variog, "modelVariog"), aes(x=dist, y=variog), color="blue") +
  geom_hline(yintercept=1,linetype="dashed")

e) Add random factor

The package nlme allows us also to include random factors. Here we add System as a random factor and test whether this would improve the model fit.

• Instead of function nlme::gls, we use the function nlme::lme.
• In nlme, random effects are specified differently from lmer::lme4 (Week6): random = ~ 1 | System.
• The correlation structure is specified exactly as with gls.

mod.lme.corExp <- nlme::lme( A ~ PatchSize + IBR, 
                             random = ~ 1 | System, data = Dianthus.df, 
                            correlation = nlme::corExp(form = ~ x + y, nugget=T),
                            method="REML")
summary(mod.lme.corExp)

## Linear mixed-effects model fit by REML
##   Data: Dianthus.df 
##           AIC     BIC   logLik
##   -0.04236554 14.1351 7.021183
## 
## Random effects:
##  Formula: ~1 | System
##         (Intercept)  Residual
## StdDev:   0.0816568 0.1947089
## 
## Correlation Structure: Exponential spatial correlation
##  Formula: ~x + y | System 
##  Parameter estimate(s):
##       range      nugget 
## 385.7379808   0.1971837 
## Fixed effects:  A ~ PatchSize + IBR 
##                Value  Std.Error DF  t-value p-value
## (Intercept) 4.084490 0.08657381 53 47.17928  0.0000
## PatchSize   0.049722 0.01860186 53  2.67296  0.0100
## IBR         0.083831 0.10438753 53  0.80307  0.4255
##  Correlation: 
##           (Intr) PtchSz
## PatchSize  0.560       
## IBR       -0.782 -0.485
## 
## Standardized Within-Group Residuals:
##          Min           Q1          Med           Q3          Max 
## -3.156087748 -0.303811942  0.006632744  0.524431829  1.893761280 
## 
## Number of Observations: 59
## Number of Groups: 4

The nature of the results did not change: PatchSize is still significant but IBR is not.

As in Week 6, we can obtain marginal (fixed effects) and conditional R-squared values (fixed + random):

MuMIn::r.squaredGLMM(mod.lme.corExp)

##            R2m       R2c
## [1,] 0.2294698 0.3447197

Now we can include this model in the model comparison from above. Notes:

• As discussed in the Week 6 video, we should fit the mixed model with maximum likelihood (method = “ML”) to test fixed effects and to compare its AIC to the other models.
• The Week 6 video followed the philosphy that random effect should only be fitted for factors with >5 levels, whereas here, we are using a factor with 4 levels.

MuMIn::model.sel(model.lm, mod.corExp, mod.corRatio, mod.lme.corExp)  

## Model selection table 
##                (Intrc)     IBR   PtchS             family class
## mod.corExp       4.044 0.08468 0.04269 gaussian(identity)   gls
## mod.corRatio     4.032 0.09704 0.04129 gaussian(identity)   gls
## model.lm         4.052 0.11340 0.04266 gaussian(identity)   gls
## mod.lme.corExp   4.084 0.08383 0.04972 gaussian(identity)   lme
##                           correlation random df logLik AICc delta weight
## mod.corExp       nlme::corExp(~x+y,T)         6  6.283  1.1  0.00  0.295
## mod.corRatio   nlme::corRatio(~x+y,T)         6  6.229  1.2  0.11  0.280
## model.lm                                      4  3.702  1.3  0.29  0.256
## mod.lme.corExp   nlme::corExp(~x+y,T)      S  7  7.021  2.2  1.10  0.170
## Models ranked by AICc(x) 
## Random terms: 
## S = '1 | System'

Questions:

• How may degrees of freedom (df) were used for the random effect?
• How can you see from the table that System was fitted as a random effect, and what method was used (REML vs. ML)?
• Was the model with the random effect ranked higher than the model without it?
• Compare the estimates of the slope coefficients for IBR between the models. How did accounting for spatial autocorrelation affect the slope coefficient, compared to model.lm? How large was the difference due to using different variogram models? And how much of a difference was related to including the random effect?
• How about the slope estimate for PatchSize?

a) Fit and compare alternative SAR models

The method errorsarlm fits a simultaneous autoregressive model (‘sar’) to the error (‘error’) term of a ‘lm’ model.

This approach is based on spatial neighbours and weights. We have already defined them in three versions of a listw object. Let’s see which one fits the data best. First, we fit the three models:

mod.sar.IBR.gab <- spatialreg::errorsarlm(A ~ PatchSize + IBR, data = Dianthus.df, 
                                 listw = listw.gab)
mod.sar.IBR.d1 <- spatialreg::errorsarlm(A ~ PatchSize + IBR, data = Dianthus.df, 
                                 listw = listw.d1)
mod.sar.IBR.d2 <- spatialreg::errorsarlm(A ~ PatchSize + IBR, data = Dianthus.df, 
                                 listw = listw.d2)

Due to some issues when using model.sel with these objects, here we manually compile AICc and delta values and sort the models by delta:

#MuMIn::model.sel(mod.lm.IBR, mod.sar.IBR.gab, mod.sar.IBR.d1, mod.sar.IBR.d2) 

Models <- list(mod.lm.IBR=mod.lm.IBR, mod.sar.IBR.gab=mod.sar.IBR.gab, 
               mod.sar.IBR.d1=mod.sar.IBR.d1, mod.sar.IBR.d2=mod.sar.IBR.d2)
data.frame(AICc = sapply(Models, MuMIn::AICc)) %>% 
  mutate(delta = AICc - min(AICc)) %>%
  arrange(delta)

##                      AICc     delta
## mod.sar.IBR.d1  -13.86736 0.0000000
## mod.sar.IBR.gab -13.29796 0.5694002
## mod.sar.IBR.d2  -11.96555 1.9018039
## mod.lm.IBR      -10.77004 3.0973135

The best model (‘mod.sar.IBR.d1’) is the one with (row-standardized) inverse-distance weights (‘listw.d1’). It is only slightly better than the model with the (row-standardized) binary weights (‘listw.gab’), whereas the nonspatial model and the one with (row-standardized) inverse squared distance weights have much less support.

b) Interpret best-fitting SAR model

Let’s have a look at the best model. With the argument Nagelkerke = TRUE, we request a pseudo R-squared.

summary(mod.sar.IBR.d1, Nagelkerke = TRUE)

## 
## Call:spatialreg::errorsarlm(formula = A ~ PatchSize + IBR, data = Dianthus.df, 
##     listw = listw.d1)
## 
## Residuals:
##       Min        1Q    Median        3Q       Max 
## -0.654613 -0.085961  0.016066  0.091687  0.389419 
## 
## Type: error 
## Coefficients: (asymptotic standard errors) 
##             Estimate Std. Error z value Pr(>|z|)
## (Intercept) 4.067066   0.077809 52.2696   <2e-16
## PatchSize   0.041416   0.018554  2.2322   0.0256
## IBR         0.090578   0.094946  0.9540   0.3401
## 
## Lambda: 0.22322, LR test value: 2.6442, p-value: 0.10393
## Asymptotic standard error: 0.13659
##     z-value: 1.6343, p-value: 0.1022
## Wald statistic: 2.6708, p-value: 0.1022
## 
## Log likelihood: 12.49972 for error model
## ML residual variance (sigma squared): 0.037595, (sigma: 0.19389)
## Nagelkerke pseudo-R-squared: 0.26613 
## Number of observations: 59 
## Number of parameters estimated: 5 
## AIC: -14.999, (AIC for lm: -14.355)

• Again, PatchSize is significant but not IBR.
• The section starting with ‘Lamba’ summarizes the fitted spatial autocorrelation term. It is not statistically significant (p-value = 0.1039 for the Likelihood Ratio test LR).

a) Default method

The new package spmoran makes this really easy. First, we create the MEM spatial eigenvectors. This implies defining neighbors and weights, but this is well hidden in the code below. The function meigen here takes the coordinates, calculates a minimum spanning tree (so that each site has at least one neighbour), and finds the maximum distance ‘h’ from the spanning tree. It then calculates neighbor weights as exp(-dij / h).

Note: if you have many sites (> 200), the function meigen_f may be used instead of meigen, it should even work for >1000 sites.

The function esf then performs the spatial filtering. Here it uses stepwise selection of MEM spatial eigenvectors using an R-squared criterion (fn = "r2").

# lm model: using truncated distance matrix (max of min spanning tree distance)
meig <- spmoran::meigen(coords=xy)

##  9/59 eigen-pairs are extracted

sfd.res <- spmoran::esf( y=Dianthus.df$A, x=Dianthus.df[,c("PatchSize", "IBR")],
                       meig=meig, fn = "r2" )

##   5/9 eigenvectors are selected

The objects created by functions ‘meigen’ and ‘esf’ contain a lot of information:

• meigW: a list returned by function ‘meigen’, with the following attributes:
  • sf: Matrix of retained spatial eigenvectors.
  • ev: Eigenvalues of retained spatial eigenvectors.
  • ev_full: All (n - 1) eigenvalues.
• sfd.res: a list returned by function ‘esf’, with the following attributes:
  • b: Table with regression results for predictors X.
  • r: Table with regression results the selected MEM spatial eigenvectors (based on step-wise eigenvector selection).
  • e: Summary statistics for the entire model.
  • vif: Variance inflation factors.
  • sf: Fitted spatially dependent component (i.e., fitted value based on significant MEM spatial eigenvectors)
  • pred: Fitted values.
  • resid: Residuals.

Let’s look at the table ‘b’ with regression results for the predictors first:

sfd.res$b

##               Estimate         SE    t_value      p_value
## (Intercept) 4.08678879 0.06759658 60.4585130 3.838659e-49
## PatchSize   0.03642554 0.01673612  2.1764625 3.417603e-02
## IBR         0.04687286 0.08602602  0.5448685 5.882188e-01

Again, PatchSize is statistically significant but not IBR.

Next, we look at the table ‘r’ with regression results for MEM spatial eigenvectors:

sfd.res$r

##       Estimate        SE   t_value     p_value
## sf6  0.5938646 0.1782765  3.331144 0.001613967
## sf1  0.3962440 0.1788408  2.215624 0.031207615
## sf7 -0.4065267 0.1895836 -2.144313 0.036795842
## sf3 -0.3474148 0.1760289 -1.973624 0.053854181
## sf5 -0.3290397 0.1757533 -1.872168 0.066922529

Five MEM spatial eigenvectors were important enough to be included in the model. Here they are ranked by their (absolute value of) slope coefficient, and thus by the strength of their association with the response variable. Eigenvector ‘sf6’ was by far the most important.

Note: some eigenvectors are included despite having a p-value > 0.05. This may have two reasons. First, the eigenvectors were selected without taking into account predictors X. Second, a different test was used in the stepwise eigenvector selection. The type of test can be specified with an argument fn (see ‘?esf’ helpfile and ‘spmoran’ tutorial).

Finally, let’s look at the summary results for the fitted model:

sfd.res$e

##                 stat
## resid_SE   0.1728755
## adjR2      0.4374576
## logLik    24.1369653
## AIC      -30.2739307
## BIC      -11.5760937

Here, ‘adjR2’ is rather high (0.437), but this includes the selected MEM spatial eigenvectors!

b) Using a custom connectivity matrix

We know already that listw.d1 fit the data well, so let’s re-run the model with our own definition of spatial weights. With the funciton ‘listw2mat’, we convert from listw format to a full connnectivity matrix.

cmat.d1    <- spdep::listw2mat( listw.d1) 
meigw  <- spmoran::meigen( cmat = cmat.d1 )

##  Note: cmat is symmetrized by ( cmat + t( cmat ) ) / 2

##  27/59 eigen-pairs are extracted

sfw.res <- spmoran::esf( y=Dianthus.df$A, x=Dianthus.df[,c("PatchSize", "IBR")],
                       meig=meigw, fn = "r2" )

##   15/27 eigenvectors are selected

sfw.res$b

##               Estimate         SE    t_value      p_value
## (Intercept) 4.05985212 0.06875531 59.0478330 2.716097e-41
## PatchSize   0.02051541 0.01679035  1.2218572 2.287404e-01
## IBR         0.06583796 0.09048375  0.7276219 4.709800e-01

tibble::as_tibble(sfw.res$r)

## # A tibble: 15 × 4
##    Estimate    SE t_value  p_value
##       <dbl> <dbl>   <dbl>    <dbl>
##  1   -0.712 0.172   -4.14 0.000166
##  2    0.511 0.166    3.08 0.00369 
##  3   -0.366 0.151   -2.43 0.0196  
##  4    0.284 0.152    1.87 0.0693  
##  5   -0.278 0.155   -1.79 0.0806  
##  6    0.236 0.167    1.41 0.166   
##  7    0.270 0.150    1.80 0.0787  
##  8    0.309 0.159    1.94 0.0594  
##  9   -0.289 0.160   -1.80 0.0786  
## 10    0.227 0.152    1.49 0.143   
## 11    0.233 0.151    1.55 0.130   
## 12    0.217 0.151    1.43 0.159   
## 13   -0.189 0.151   -1.25 0.217   
## 14    0.196 0.152    1.29 0.204   
## 15    0.166 0.161    1.03 0.309

sfw.res$e

##                 stat
## resid_SE   0.1493144
## adjR2      0.5803455
## logLik    39.2199714
## AIC      -40.4399428
## BIC       -0.9667313

Note: the messages tell us that ‘cmat’ has been made symmetric before analysis, that 27 out of 59 MEM spatial eigenvector (and their eigenvalues, hence ‘pairs’) were retained initially and subjected to stepwise selection, which then returned 15 statistically significant MEM eigenvectors that were included in the regression model with the predictor variables X (PatchSize and IBR).

Questions:

• Does this model fit the data better? Look for a lower AIC. In addition, you can compare the adjusted R-squared.
• What could cause a difference in model performance?
• Does this affect the results for PatchSize and IBR? Compare both parameter estimates and p-value between two two models.

c) Plot spatial eigenvectors

So far, we have treated the MEM spatial eigenvectors as a black box. What kind of patterns do they represent?

First, we plot all the selected (significant) eigenvectors. A convenient way to do so is converting to a SpatialPointsDataFrame and then use the function ‘spplot’.

Here we need tweak the column names of the eigenvectors, which are called “X1”, “X2” etc., to match the row names of table ‘sf.res$r’, which uses “sf1”, “sf2” etc.

• In the first line, we create a data frame ‘SF’ that combines the coordinates from ‘xy’ with the eigenvectors from ‘meigw’. By specifying sf=meigw$sf, the column names will be prefaced by ‘sf.’, so that “X1” becomes “sf.1”.
• With the function gsub, we substitute “.” by an empty vector ““, i.e., we remove the period from the name, so that”sf.1” becomes “sf1”. This is a bit tricky because of the special meanings of ‘.’: we need to escape (here we use “[.]”) to indicate that ‘.’ should be interpreted and matched literally as a period symbol, not as a wildcard that matches any character. Function sub substitutes the first occurrence of the symbol, whereas gsub substitutes all occurrences.

SF <- data.frame(xy, sf=meigw$sf)
names(SF) <- gsub("[.]", "", names(SF))
sp::coordinates(SF) <- ~ x + y
sp::spplot(SF, row.names(sfw.res$r), colorkey = TRUE)

The panel starts at the bottom left, so that the most important spatial eigenvector (sf6) is plotted at the bottom left, the second most important (sf22) second from left, etc.

The smallest numbers are patterns with the largest spatial scale (sf1), which here shows a gradient from East to West. The most important eigenvector (sf6) shows a finer-scale pattern with the highest values (yellow) in the center, lowest values East and West, and intermediate value North and South.

However, these patterns individually are not meaningful. More importantly, we can plot the total spatial component in the response as a weighted sum of these component patterns, where the weights correspond to the regression coefficients in table ‘sf.res$r’. Here we create a panel with two plots, the modeled spatial components ‘sf’ on the left and the response ‘A’ on the right (the mean has been removed to make values comparable).

SF@data$sf <- sfw.res$sf
SF@data$A <- scale(Dianthus.df$A, scale = FALSE)
sp::spplot(SF, c("sf", "A"), colorkey = TRUE)

Obviously, a big part of the variation in allelic richness is already captured by ‘sf’. In essence the model then tries to explain the difference between these two sets of values by the predictors “PatchSize” and “IBR”.

Let’s quantify the correlation of this spatial component with allelic richness, and compare the correlation between the two models:

cor(Dianthus.df$A, data.frame(sfd=sfd.res$sf, sfw=sfw.res$sf))

##            sfd       sfw
## [1,] 0.6252594 0.8054356

With the default method (defining neighbors based on a distance cut-off), the spatial component modeled by the significant MEM spatial eigenvectors showed a correlation of 0.625 with the response variable. Using a Gabriel graph with inverse distance weights increased this correlation to 0.772.

This means that the spatial eigenvectors derived from the Gabriel graph were more effective at capturing the spatial variation in allelic richness than the default method. This spatial component is then controlled for when assessing the relationship between allelic richness and the predictors (PatchSize and IBR).

d) Random effect model

The previous model selected 15 MEM spatial eigenvectors, and thus fitted 15 additional models. Just like the random effects for family and population in Week 6 lab, we can save a few parameters here by fitting the set of MEM eigenvectors as a random effect. This is done by the function ‘resf’.

sfr.res <- spmoran::resf( y=Dianthus.df$A, x=Dianthus.df[,c("PatchSize", "IBR")], 
               meig = meigw, method = "reml" ) 

## Note: The model is nearly singular. Consider simplifying the model

sfr.res$b

##               Estimate         SE   t_value    p_value
## (Intercept) 4.06902708 0.06645938 61.225772 0.00000000
## PatchSize   0.04247164 0.01652051  2.570843 0.01344639
## IBR         0.08489102 0.08467569  1.002543 0.32132728

tibble::as_tibble(sfr.res$r)

## # A tibble: 27 × 1
##      value
##      <dbl>
##  1 -0.191 
##  2 -0.0314
##  3 -0.0601
##  4 -0.124 
##  5  0.0640
##  6 -0.312 
##  7 -0.0576
##  8 -0.113 
##  9  0.128 
## 10  0.105 
## # … with 17 more rows

sfr.res$e

##                    stat
## resid_SE     0.15643276
## adjR2(cond)  0.50394641
## rlogLik      5.99141269
## AIC          0.01717462
## BIC         12.48239928

sfr.res$s

##                      (Intercept)
## random_SE              0.1785609
## Moran.I/max(Moran.I)   0.7742903

As in Week 6 lab, the conditional R-squared is the variance explained by the fixed effects (PatchSize and IBR) and the random effects (significant MEM spatial eigenvectors) together. It is adjusted for the number of effects that were estimated.

Note: we can’t compare AIC with the previous models, as the model was fitted with ‘reml’.

We get an additional output ‘sfr.res$s’ with two variance parameters:

• spcomp_SE: standard error of the spatial component.
• spcomp_Moran.I/max(Moran.I): Moran’s I of the spatial component, rescaled by the maximum possible value. From the help file: “Based on Griffith (2003), the scaled Moran’I value is interpretable as follows: 0.25-0.50:weak; 0.50-0.70:moderate; 0.70-0.90:strong; 0.90-1.00:marked.”

Model with PatchSize and IBR

We fit the model with ‘resf_vc’.

rv_res <- spmoran::resf_vc( y=Dianthus.df$A, 
                            x = Dianthus.df[,c("PatchSize", "IBR")], 
                            xconst = NULL, meig = meigw, method = "reml")

## [1] "-------  Iteration 1  -------"
## [1] "1/3"
## [1] "2/3"
## [1] "3/3"
## [1] "BIC: 10.465"
## [1] "-------  Iteration 2  -------"
## [1] "1/3"
## [1] "2/3"
## [1] "3/3"
## [1] "BIC: 10.37"
## [1] "-------  Iteration 3  -------"
## [1] "1/3"
## [1] "2/3"
## [1] "3/3"
## [1] "BIC: 10.37"
## [1] "-------  Iteration 4  -------"
## [1] "1/3"
## [1] "2/3"
## [1] "3/3"
## [1] "BIC: 10.37"

## Note: The model is nearly singular. Consider simplifying the model

Instead of one slope estimate for each predictor, we now get a different estimate for each combination of parameter and site (sounds like overfitting?). Here’s a summary of the distbiution of these estimates.

summary( rv_res$b_vc ) 

##   (Intercept)      PatchSize            IBR         
##  Min.   :3.877   Min.   :0.04717   Min.   :0.06883  
##  1st Qu.:4.049   1st Qu.:0.04717   1st Qu.:0.06883  
##  Median :4.106   Median :0.04717   Median :0.06883  
##  Mean   :4.083   Mean   :0.04717   Mean   :0.06883  
##  3rd Qu.:4.128   3rd Qu.:0.04717   3rd Qu.:0.06883  
##  Max.   :4.189   Max.   :0.04717   Max.   :0.06883

The slope estimate for PatchSize varied between 0 and 0.098, with a mean of 0.046. The slope estimate for the ‘IBR’ term varied between -0.81 and 0.34, with a mean very close to 0! That is an astounding range of variation. Keep in mind that we really expect a positive relationship, there is no biological explanation for a ngeative relationship.

Here is a similar summary of the p-values:

summary( rv_res$p_vc )

##   (Intercept)   PatchSize             IBR        
##  Min.   :0    Min.   :0.005485   Min.   :0.4135  
##  1st Qu.:0    1st Qu.:0.005485   1st Qu.:0.4135  
##  Median :0    Median :0.005485   Median :0.4135  
##  Mean   :0    Mean   :0.005485   Mean   :0.4135  
##  3rd Qu.:0    3rd Qu.:0.005485   3rd Qu.:0.4135  
##  Max.   :0    Max.   :0.005485   Max.   :0.4135

For both variables, most sites do not show a significant effect (i.e., only few sites show a p-value < 0.05).

We could print these results by site (type rv_res$b_vc or rv_res$p_vc). Even better, we can plot them in space. We start with combining the data (‘Dianthus.df’) and the results into one data frame ‘Results’. By specifying b=rv_res$b_vc and p=rv_res$p_vc, R will create column names that start with ‘b’ or ‘p’, respectively.

Result <- data.frame(Dianthus.df, b=rv_res$b_vc, p=rv_res$p_vc)
names(Result)

##  [1] "A"             "IBD"           "IBR"           "PatchSize"    
##  [5] "System"        "Longitude"     "Latitude"      "x"            
##  [9] "y"             "b..Intercept." "b.PatchSize"   "b.IBR"        
## [13] "p..Intercept." "p.PatchSize"   "p.IBR"

Let’s start with PatchSize. Here, we first plot PatchSize in space, with symbol size as a function of patch size. In a second plot, we color sites by statistical significance and the size of the symbols represents the parameter estimate of the regression slope coefficient for Patch Size. The layer ‘coord_fixed’ keeps controls the aspect ratio between x- and y-axes.

require(ggplot2)
ggplot(as.data.frame(Result), aes(x, y, size=PatchSize)) +
  geom_point(color="darkblue") + coord_fixed()
ggplot(as.data.frame(Result), aes(x, y, col=p.PatchSize < 0.05, size=b.PatchSize)) +
  geom_point() + coord_fixed()

Let’s do the same for ‘IBR’:

require(ggplot2)
ggplot(as.data.frame(Result), aes(x, y, size=IBR)) +
  geom_point(color="darkgreen") + coord_fixed()
ggplot(as.data.frame(Result), aes(x, y, col=p.IBR < 0.05, size=b.IBR)) +
  geom_point() + coord_fixed()

• The very small dots in the first map are the ungrazed patches.
• From the second map, it looks like the significant values were the one with negative slope estimates, for which we don’t have a biological interpretation.

Model with IBR only

Keep in mind that ‘IBR’ and ‘PatchSize’ showed a strong correlation. The parameter estimates could therefore depend quite a bit on the other variables. To help with the interpretation, let’s repeat the last analysis just with ‘IBR’, without ‘PatchSize’.

rv_res <- spmoran::resf_vc( y=Dianthus.df$A, 
                            x = Dianthus.df[,c("IBR")], 
                            xconst = NULL, meig = meigw, method = "reml")

## [1] "-------  Iteration 1  -------"
## [1] "1/2"
## [1] "2/2"
## [1] "BIC: 5.181"
## [1] "-------  Iteration 2  -------"
## [1] "1/2"
## [1] "2/2"
## [1] "BIC: 5.226"
## [1] "-------  Iteration 3  -------"
## [1] "1/2"
## [1] "2/2"
## [1] "BIC: 5.226"
## [1] "-------  Iteration 4  -------"
## [1] "1/2"
## [1] "2/2"
## [1] "BIC: 5.226"

## Note: The model is nearly singular. Consider simplifying the model

summary( rv_res$b_vc ) 

##   (Intercept)          V1        
##  Min.   :3.737   Min.   :0.2264  
##  1st Qu.:3.902   1st Qu.:0.2264  
##  Median :3.961   Median :0.2264  
##  Mean   :3.941   Mean   :0.2264  
##  3rd Qu.:3.994   3rd Qu.:0.2264  
##  Max.   :4.064   Max.   :0.2264

Now the range of slope estimates is smaller, most sites have a positive estimate, and the mean is approx. 0.21.

summary( rv_res$p_vc ) 

##   (Intercept)       V1          
##  Min.   :0    Min.   :0.001875  
##  1st Qu.:0    1st Qu.:0.001875  
##  Median :0    Median :0.001875  
##  Mean   :0    Mean   :0.001875  
##  3rd Qu.:0    3rd Qu.:0.001875  
##  Max.   :0    Max.   :0.001875

Also, a larger proportion of sites nows has p-values < 0.05.

Let’s plot the results onto a gray-scale, stamen terrain map to facilitate interpretation. Note: here the zoom level zoom = 12 covers the entire study area, whereas the default value would actually cut off a large number of sites. We use the argument force=TRUE to force the map to be downloaded again (otherwise the argument color="bw" may not have an effect if we already downloaded the terrain map in color).

Result <- data.frame(Dianthus.df, b=rv_res$b_vc, p=rv_res$p_vc, resid=rv_res$resid)
names(Result)

##  [1] "A"             "IBD"           "IBR"           "PatchSize"    
##  [5] "System"        "Longitude"     "Latitude"      "x"            
##  [9] "y"             "b..Intercept." "b.V1"          "p..Intercept."
## [13] "p.V1"          "resid"

ggmap::ggmap(get_stamenmap(bbox, maptype = "terrain", color="bw", force=TRUE, zoom=12)) + 
   geom_point(aes(x =  Longitude, y = Latitude, col=p.V1 < 0.05, size=b.V1), data = Result) 

This is a very different map of results!

• Most sites now show significant effects.
• The sites with larger positive estimates show significant effects, whereas those with small or negative estimates show non-significant effects.
• There are 3 - 4 clusters of sites where the IBR models is not effective at explaining variation in allelic richness: in the very East, in the South-East, and one area in the South-West.
• Knowing the study area, these are distinct regions (e.g. valleys) that may suggest further biological explanations.